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A359846
a(n) = (((5 - (n mod 2))*10^(3 + n*(9/2) - (n mod 2)*(5/2)))^2 + 2)/81.
0
308642, 1975308642, 308641975308641975308642, 1975308641975308641975308642, 308641975308641975308641975308641975308642, 1975308641975308641975308641975308641975308642, 308641975308641975308641975308641975308641975308641975308642
OFFSET
0,1
COMMENTS
Also numbers of the form ((d*10^k)^2 + 2)/9^2 that are not squares, where d is a single-digit number.
The square roots of these numbers show runs of equal digits, see the link to Schizophrenic numbers.
FORMULA
G.f.: (2/81)*(1/(1-x)+6249960/(1+1000000000*x)+6250040/(1-1000000000*x)).
a(n) = a(n-1) + 10^18*a(n-2) - 10^18*a(n-3).
a(2*n) = (25*10^(6 + 18*n) + 2)/81.
a(2*n + 1) = (16*10^(10 + 18*n) + 2)/81.
We use in the next formulas a special notation for real numbers where (x) after a digit denotes a run of length x for this digit. Example: 3(4).2(3) is 3333.2222 .
sqrt(a(2*n)) = 5(3+9*n).5(4+9*n)7(8+18*n)3(7+18*n)5(1)1(6+18*n)0(1)2(7+18*n)7(1)1(1)9(5+18*n)7(1)0(1)1(1)3(4+18*n)... .
sqrt(a(2*n+1)) = 4(5+9*n).4(6+9*n)7(1)2(10+18*n)1(1)3(1)5(1)4(1)1(1)6(6+18*n)7(1)2(1)0(1)9(1)2(1)0(1)1(1)3(1)8(4+18*n)... .
sqrt(1/a(2*n)) = 0.0(2+9*n)1(1)7(1)9(6+18*n)2(1)8(1)0(5+18*n)... .
sqrt(1/a(2*n+1)) = 0.0(4+9*n)2(2)4(1)9(8+18*n)8(1)5(1)9(1)3(1)7(1)5(1)0(5+18*n)... .
sqrt(a(2*n)-(2/81)) = 10^(4+9*n)/18.
sqrt(a(2*n+1)-(2/81)) = 10^(7+9*n)/225.
EXAMPLE
a(0) = 308642 and sqrt(a(0)) = 555.555577777777333333351111110222222271999997013333521... .
a(1) = 1975308642 and sqrt(a(1)) = 44444.4444447222222222213541666666720920138888465033637... .
PROG
(PARI) a(n) = (((5-(n%2))*10^(3+n*(9/2)-(n%2)*(5/2)))^2+2)/81
CROSSREFS
Cf. A014824.
Sequence in context: A236818 A234188 A226364 * A206276 A251304 A127491
KEYWORD
nonn,easy,base
AUTHOR
Thomas Scheuerle, Jan 15 2023
STATUS
approved